Emden-Fowler Equation and Inverse Analysis of Simple Flows through Variable Permeability Porous Layers

Size: px

Start display at page:

Download "Emden-Fowler Equation and Inverse Analysis of Simple Flows through Variable Permeability Porous Layers"

Dwayne Freeman
5 years ago
Views:

1 4 Int. J. Open Problems Compt. Math., Vol. 0, No., June 07 ISSN ; Copyright ICSRS Publication, 07 Emden-Fowler Equation and Inverse Analysis of Simple Flows through Variable Permeability Porous Layers M.H. Hamdan and M.S. Abu Zaytoon Department of Mathematics Statistics, University of New Brunswic, P.O. Box 5050, Saint John, New Brunswic, Canada EL 4L5 Corresponding Author Abstract In this wor we introduce an inverse method to analyze simple flows through variable permeability porous layers. Assuming that the velocity distribution is given, or specified as a function of the permeability, the governing equation is solved for the permeability distribution, then the velocity function is then recovered. Poiseuille-type flow involving Brinman s equation is considered together with other flow problems involving coupled parallel flow through composite layers. In case of flow through a Brinman-Forchheimer layer over a Darcy layer, the governing equation was transformed into an Emden-Fowler equation whose solution provides a method for determining Beavers and Joseph slip parameter. Keywords: Brinman-Forchheimer, Variable Permeability, Emden-Fowler Equation, Slip Parameter. Introduction Many excellent reviews of flow through and over porous layers are available in the literature, (cf. [,5,8,]) and discuss all aspects of the flow phenomena, including the various flow models and their validity, applications, solutions and analysis of the model equation. Modelling fluid flow through porous layers with variable permeability can be argued to be more realistic in representing natural phenomena as compared to flow through constant permeability layers [], and finds applications in industry and nature, and has implications in the analysis of the transition layer [9]. However, analysis of this type of flow presents challenges at

2 5 M.H. Hamdan and M.S. Abu Zaytoon more than one front, including modelling permeability variations and solutions to the resulting governing equations. The momentum equation in this type of unidirectional flow involves two functions to be solved for: velocity and permeability. If permeability distribution is specified then velocity can be solved for. A number of variable permeability models have been introduced and successfully analyzed in the literature [], and are based mainly on specifying the permeability distribution and solving the resulting governing equation for the velocity function (cf. [,6] and the references therein). In a recent article however, [6], the authors used a non-dimensionalizing procedure that resulted in a variable permeability distribution, for Brinman s equation, that is tied to the velocity distribution. We capitalize on this idea in the current wor where we consider simple flows in which the velocity is pre-defined in terms of the variable permeability in order to obtain a permeability equation that can be solved for the permeability distribution and subsequently the velocity is recovered. We illustrate this inverse approach by considering five flow configurations that involve Brinman s equation and the Brinman-Forchheimer equation and obtain a permeability equation that is easily solved. In case of the Brinman-Forchheimer equation, we reduce the permeability equation to the Emden-Fowler equation, [4,0], whose solution is readily available. Inverse analysis in this case can provide information on the slip parameter of the Beavers and Jospeh condition, [3]. Problem Formulation The steady flow of a viscous, incompressible fluid, in the absence of body forces, through porous media is governed by the continuity equation, namely v 0 () and the following general momentum equation that incorporates Darcian and non-darcian ects, nown as the Brinman- Forchheimer equation: C f ( v v) p v v v v () wherein p is the pressure, is the permeability, v is the velocity vector, is the fluid density, C f is the Forchheimer drag coicient, is the base fluid viscosity, and is the ective viscosity of the fluid saturating the porous medium. For parallel, unidirectional flow through a porous layer with variable permeability, governing equations () and () reduce to: d u C f u u (3) dy ( where u u( is the tangential velocity component.

3 Emden-Fowler Equation and Inverse Analysis 6 If the permeability distribution ( is given, equation (3) can be solved for the velocity distribution u (. On the other hand, if the velocity distribution u ( is given as a function of the permeability, equation (3) gives an equation that the permeability has to satisfy. Once the permeability distribution is solved for, the associated velocity distribution can be obtained. Thus, if u( f ( ( ) then (3) taes the form C f ( f ( ( ) ( f ( ( ) f ( ( ) [ f ( ( )] ( (4). (5) Clearly, with the nowledge of either ( or f ( ( ), equation (5) can be solved for the other function. In what follows we consider simple flows in which f ( ( ) is given. 3 Simple Flows 3. Flow through a variable-permeability Brinman porous layer between parallel plates Consider the unidirectional flow through a Brinman porous layer between solid, parallel plates located at y 0 and y h. On the solid walls, the no-slip velocity condition u ( 0) u( 0 and the non-penetrability condition ( 0) ( 0 are imposed. We assume that the channel is squeezed to the point that permeability is continuously varying. The flow is governed by Brinman s equation which taes the following form for the configuration in Fig., obtained by setting C to zero in equation (3): f d u u. (6) dy ( y y=h y=0 Brinman porous layer Flow direction Fig.. Representative setch of flow between parallel plates Assuming that equation (4) is valid for this flow, equation (5) reduces to:

4 7 M.H. Hamdan and M.S. Abu Zaytoon ( f ( ( ) ( f ( ( ) f ( ( ). (7) ( Equation (7) can be solved for ( if given the form of f ( ( ). For the sae of illustration, assume that u( f ( ( ) A (8) where A is a constant. Using (8) in (7), we obtain (. A (9) Solution to (9) satisfying ( 0) ( 0 is given by ( [ A ]( y h (0) and velocity distribution (8) taes the form A u( [ ]( y h. () Equations (0) and () give maximum permeability, y h / respectively as max, and maximum velocity, u max, at max h h ( ) [ ] () 8 A u h h ( A 8 A max u ) A[ ] max. (3) We note that equation (0) is the dimensional form of the permeability function obtained in [6] through non-dimensionalizing. 3. Flow through a variable-permeability finite Brinman porous layer over a semi-infinite Darcy layer Consider the unidirectional flow through a Brinman porous layer between y 0 and y h, over a semi-infinite Darcy layer of constant permeability, depicted in Fig.. The flow is

5 Emden-Fowler Equation and Inverse Analysis 8 governed by Brinman s equation (6) in the Brinman layer and in the Darcy layer by Darcy s law, of the form u D (4) where u is the Darcy velocity, D is the constant permeability in the Darcy layer and 0 is the common driving pressure gradient. On the solid wall at y h, the no-slip velocity condition u ( 0 and the non-penetrability condition ( 0 are imposed. At the interface, y 0, the following Beavers and Joseph condition [3] is valid y y=h y=0 Brinman porous layer Darcy semi-infinite layer Flow direction Fig.. Representative setch of flow through a finite Brinman layer over a semi-infinite Darcy layer u where ( u B ud ) at y 0 (5) u u( 0 B ). (6) Assuming that (8) is valid, then using (8) in (6) results in (. A (7) Solution to (7) taes the form y ( [ ] c y c. (8) A where c,c are arbitrary constants. Velocity in the Brinman layer is thus given by

6 9 M.H. Hamdan and M.S. Abu Zaytoon y u( A A[ ] c Ay c A and its derivative is given by A (9) u( A[ ] y c A. (0) A Using u ( 0, in (9), we obtain h ] ch c [ A 0. () From (6) and (9) we obtain u( 0) ub c A () and from (0) we obtain ( 0) c A. u (3) Using (4), () and (3) in (5) we obtain c c. A (4) Equations () and (4) are solved for the arbitrary constants c and c and yield h h Ah c ( ) (5) A[ h ] h Ah c ( ). (6) A[ h With c and c determined, the flow quantities are described as follows.

7 Emden-Fowler Equation and Inverse Analysis 30 ( [ A h A[ h A u( [ h [ h y ] A[ h h Ah ( ) ] y ] [ h h Ah ( ) ] h ( h ( Ah ) y. (7) Ah ) y. (8) u(0) [ h h ( Ah ) (9) u B h h Ah ( ). (30) [ h ] 3.3 Shear-driven flow in a semi-infinite Brinman-Forchheimer porous layer Consider the flow through a semi-infinite porous layer, shown in Fig. 3. The flow is generated by an applied pressure gradient and a moving plate, with velocity U, located at y h. The flow is governed by equation (3), valid on the interval h y. Velocity conditions to be satisfied are: u( U ; u ( ) 0; u ( ) 0. (3) y y=h Semi-infinite porous layer Plate moving with velocity U Fig. 3. Representative setch of moving plate In order to satisfy (4), (5) and (3) we assume that

8 3 M.H. Hamdan and M.S. Abu Zaytoon u f ( ) (. Equation (5) thus reduces to ( 3 / (3) (33) where C f. (34) Comparing (33) with the Emden-Fowler equation, [4,0], namely ( y n m (35) whose solution is given by y ( n)/(m ) (36) where ( n )( n m ) /( m) [ ] (37) ( m ) 3 we see that n 0 and m, and solution to (33) taes the form 0 4 ( ) y. (38) Velocity distribution thus taes the form 0 4 u ( ( ) y (39) with u ( ) y. (40) Using (39), we obtain velocity of the moving plate as u( U 0 ( ) h 4. (4)

9 Emden-Fowler Equation and Inverse Analysis 3 Conditions (3) are thus satisfied, as can be seen from (39) and (40), wherein u ( ) u( ) 0. Furthermore, equation (4) shows the parameters that the velocity of the moving plate depends on in order to have the permeability distribution given by (38). 3.4 Flow through a Brinman-Forchheimer layer over a Darcy layer Consider the coupled, parallel flow through a variable permeability Brinman- Forchheimer porous layer of semi-infinite extent ( h y ) over a constant permeability Darcy porous layer of semi-infinite extent ( y h ), shown in Fig. 4. y y=h Brinman-Forchheimer semi-infinite porous layer Darcy semi-infinite porous layer Fig. 4. Representative setch of coupled parallel flow The assumingly sharp interface between the layers is located at y h. In the Brinman- Forchheimer layer, the flow is governed by equation (3), and in the Darcy layer the flow is governed by Darcy s law, written as: u D (4) where u is the Darcy velocity and D is the constant permeability in the Darcy layer. The flow in both layers is driven by a common constant pressure gradient, 0. At the interface, y h, Beavers and Joseph condition [3,7] is assumed to be valid, namely u ( u u B D ) at y h (43) where is a slip parameter, u B u( y h ) is the velocity at the interface obtained from solution to (3). Permeability and velocity distributions in the Brinman-Forchheimer layer are obtained from solution to equation (3) and are given by () and (), from which we obtain 0 u B u( ( ) h 4 (44)

10 33 M.H. Hamdan and M.S. Abu Zaytoon 4 0 u( ( ) h 5. (45) Upon using (4), (44) and (45) in (43) we obtain the following expression for the slip parameter : 4(0). 4 h[ h (0) ] (46) Expression (46) indicates that the Beavers and Joseph slip parameter depends on the Darcy constant permeability, viscosity of the base fluid, ective viscosity of the fluid saturating the porous layer, density of the fluid, the pressure gradient, location of the interface, thicness of the porous layer and the Forchheimer drag coicient. In order to further illustrate dependence of the slip parameter on thicness of the porous layer we consider the situation in the next section. 3.5 Flow through a Brinman-Forchheimer layer sandwiched between two Darcy layers An interesting variation of the above problem of flow over a Darcy layer is the flow through a Brinman-Forchheimer layer of variable permeability that is bounded from above and below by two semi-infinite Darcy layers of constant permeability, as shown in Fig. 5. y y=h y=h Darcy semi-infinite porous layer Brinman-Forchheimer porous layer Darcy semi-infinite porous layers Fig. 5. Representative setch of a Brinman-Forchheimer porous core The Brinman-Forchheimer layer spans h y h while the Darcy layers span y h and h y. Equations governing the flow through the given configuration are equation (3) in the Brinman-Forchheimer layer, equation (4) in the lower Darcy layer, and u D (47)

11 Emden-Fowler Equation and Inverse Analysis 34 in the upper Darcy layer. Conditions at the interfaces between layers are given by u and u ( u u B D ) ( u u B D ) at at y h, (48) y h. (49) In equations (47)-(49), and are the slip parameters associated with the lower and upper interfaces, respectively, u B and u B are the lower and upper interfacial velocities, respectively, u D and u D are the Darcy velocities in the lower and upper Darcy layers, respectively, and and are the constant permeabilities in the lower and upper Darcy layers, respectively. Permeability and velocity distributions in the Brinman-Forchheimer layer are obtained from solution to equation (3) and are given by (38) and (39), which yield the following expressions for the lower and upper interfacial velocities, respectively: 0 u B u( ( ) h 4 (50) 0 4 u B u( ( ) ( and the following shear stress expressions at the lower and upper interfaces, respectively: 4 0 u( ( ) h 5 (5) (5) 4 u( 0 ( ) ( 5. (53) Upon using (4), (47) and (50) to (53) in (48) and (49), we can solve for the following expression for the slip parameters and : 4(0). (54) 4 h[ h (0) ] 4(0). (55) 4 ([ ( (0) ]

12 35 M.H. Hamdan and M.S. Abu Zaytoon It is clear from (54) and (55) that the slip parameters depend on fluid and medium properties, and on the location of the interfaces and the thicness of the Brinman-Forchheimer porous layer. 4 Conclusion In this wor we introduced an inverse method to analyze simple, unidirectional flows in variable permeability porous layers. Rather than specifying a permeability distribution and then solving the momentum equation for the velocity, we imposed a velocity distribution that is a function of the permeability and solved the momentum equation for the necessary permeability distribution. We analyzed five situations that involve simple flows through porous layers where the flows were governed by Brinman s equation and the Brinman-Forchheimer equation. In the latter case, the governing equation was transformed into an Emden-Fowler equation whose solution is well-documented in the literature. The analysis in this wor provided some insights into the determination of the Beavers and Joseph slip parameter. 5 Open Problem In most of the inverse analysis above, we relied on the particular solution of the Emden- Fowler equation to provide a variable permeability distribution that satisfies a prescribed velocity distribution. When the flow domain is composed of a Brinman-Forchheimer variable permeability porous layer that is bounded by a lower and an upper solid wall on which no-slip and no-penetration conditions are imposed, the Brinman-Forchheimer equation is reduced to the Emden-Fowler equation (33). Particular solution to this equation does not satisfy the no-slip, nopenetration conditions on the solid boundary. There is a need to construct a solution that satisfies this Poiseuille-type flow. References [] M.S. Abu Zaytoon, T.L. Alderson and M. H. Hamdan, Flow through variable permeability composite porous layers, Gen. Math. Notes, 33(), 06, [] B. Alazmi and K. Vafai, Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer, International Journal of Heat and Mass Transfer, 44, 00, [3] G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally permeable wall, Journal of Fluid Mechanics, 30, 967, [4] O.P. Bhutani and K. Vijayaumar, On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance, J. Austral. Math. Soc. Ser. B 3, 99, [5] R.A. Ford, M.S. Abu Zaytoon and M.H. Hamdan, Simulation of flow through layered porous media, IOSR Journal of Engineering, 6(6), 06, pp [6] M.H. Hamdan and M.T. Kamel, Flow through variable permeability porous layers, Adv.

13 Emden-Fowler Equation and Inverse Analysis 36 Theor. Appl. Mech., 4, 0, [7] D.A. Nield, The Beavers Joseph boundary condition and related matters: A historical and critical note, Transport in Porous Media, 78, 009, [8] D.A. Nield and A. Bejan, Convection in Porous Media, 5 th ed. Springer, (07). [9] D.A. Nield and A.V. Kuznetsov, The ect of a transition layer between a fluid and a porous medium: shear flow in a channel, Transport in Porous Media, 78, 009, [0] A.D. Polyanin and V.F. Zaitsev, Handboo of Exact Solutions for Ordinary Differential Equations, nd ed., Chapman and Hall/CRC Press, (003). [] K. Vafai and R. Thiyagaraja, Analysis of flow and heat transfer at the interface region of a porous medium, Int. J. Heat Mass Transfer, 30(7), 987,

A Permeability Function for Brinkman s Equation

Proceedings of the th WSEAS Int. Conf. on MATHEMATICAL METHOS, COMPUTATIONAL TECHNIQUES AN INTELLIGENT SYSTEMS A Permeability Function for Brinman s Equation M.H. HAMAN, M.T. KAMEL, and H.I. SIYYAM 3,